Ellipse Formula

The Formula

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
Foci: c^2 = a^2 - b^2 (where a > b). Eccentricity: e = \frac{c}{a} (with 0 \leq e < 1).

When to use: Imagine pinning two ends of a loose string to a board (these are the foci), then tracing a curve with a pencil keeping the string taut. The resulting oval shape is an ellipse. A circle is just a special ellipse where both foci coincide.

Quick Example

\frac{x^2}{25} + \frac{y^2}{9} = 1 has center (0,0), semi-major axis a = 5 (horizontal), semi-minor axis b = 3 (vertical). Foci at (\pm 4, 0) since c = \sqrt{25 - 9} = 4.

Notation

a = semi-major axis (longer), b = semi-minor axis (shorter), c = distance from center to each focus.

What This Formula Means

The set of all points in a plane where the sum of the distances to two fixed points (foci) is constant. Standard form: \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1.

Imagine pinning two ends of a loose string to a board (these are the foci), then tracing a curve with a pencil keeping the string taut. The resulting oval shape is an ellipse. A circle is just a special ellipse where both foci coincide.

Formal View

\{(x,y) \mid d((x,y), F_1) + d((x,y), F_2) = 2a\}; standard form \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 with c^2 = a^2 - b^2, eccentricity e = \frac{c}{a} < 1

Worked Examples

Example 1

easy
Find the lengths of the semi-major and semi-minor axes of the ellipse \frac{x^2}{25} + \frac{y^2}{9} = 1.

Solution

  1. 1
    The standard form is \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 where a > b > 0.
  2. 2
    Here a^2 = 25 and b^2 = 9, so a = 5 and b = 3.
  3. 3
    The semi-major axis has length a = 5 (along the x-axis) and the semi-minor axis has length b = 3 (along the y-axis).

Answer

a = 5 \text{ (semi-major)}, \quad b = 3 \text{ (semi-minor)}
An ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 has semi-major axis a (the larger denominator's square root) and semi-minor axis b. The major axis lies along whichever variable has the larger denominator.

Example 2

medium
Find the foci of the ellipse \frac{x^2}{16} + \frac{y^2}{25} = 1.

Common Mistakes

  • Confusing which denominator is a^2: a is ALWAYS the larger value. If the larger denominator is under y, the ellipse is taller than it is wide.
  • Using c^2 = a^2 + b^2 (hyperbola formula) instead of c^2 = a^2 - b^2 (ellipse formula). For ellipses, c < a; for hyperbolas, c > a.
  • Forgetting to take square roots: if a^2 = 25, then a = 5, not 25. The semi-axis lengths are a and b, not a^2 and b^2.

Why This Formula Matters

Planetary orbits are ellipses (Kepler's first law). Ellipses appear in optics (whispering galleries), satellite orbits, medical imaging (lithotripsy), and architecture.

Frequently Asked Questions

What is the Ellipse formula?

The set of all points in a plane where the sum of the distances to two fixed points (foci) is constant. Standard form: \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1.

How do you use the Ellipse formula?

Imagine pinning two ends of a loose string to a board (these are the foci), then tracing a curve with a pencil keeping the string taut. The resulting oval shape is an ellipse. A circle is just a special ellipse where both foci coincide.

What do the symbols mean in the Ellipse formula?

a = semi-major axis (longer), b = semi-minor axis (shorter), c = distance from center to each focus.

Why is the Ellipse formula important in Math?

Planetary orbits are ellipses (Kepler's first law). Ellipses appear in optics (whispering galleries), satellite orbits, medical imaging (lithotripsy), and architecture.

What do students get wrong about Ellipse?

The larger denominator is always a^2, regardless of whether it's under x or y. If a^2 is under y, the major axis is vertical.

What should I learn before the Ellipse formula?

Before studying the Ellipse formula, you should understand: equation of circle.

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